International Workshop on Quantum LDPC Codes

Spatially coupled LDPC were introduced by Felström and Zigangirov in 1999. They might be viewed in the following way, take several several instances of a certain LDPC code family, arrange them in a row and then mix the edges of the codes randomly among neighboring layers. Moreover fix the bits of the first and last layers to zero. It has soon been found out that iterative decoding behaves much better for this code than for the original LDPC code.

If one's goal is large-scale quantum computation, ultimately one wishes to minimize the amount of time, number of qubits, and qubit connectivity required to outperform a classical system, all while assuming some physically reasonable gate error rate. We present two examples of such an overhead study, focusing on the surface code with and without long-range interactions.

All examples of quantum LDPC codes known to this date suffer from a poor distance scaling limited by the square-root of the code length. This is in a sharp contrast with the classical case where good LDPC codes are known that combine constant encoding rate and linear distance. In this talk I will describe the first family of good quantum "almost LDPC" codes. The new codes have a constant encoding rate, linear distance, and stabilizers acting on at most square root of n qubits, where n is the code length.

I will describe a new class of topological quantum error correcting codes with surprising features. The constructions is based on color codes: it preserves their unusual transversality properties but removes important drawbacks. In 3D, the new codes allow the effectively transversal implementation of a universal set of gates by gauge fixing, while error-dectecting measurements involve only 4 or 6 qubits. Furthermore, they do not require multiple rounds of error detection to achieve fault-tolerance.

This talk is divided into two parts. In the first part, I discuss a scheme of fault-tolerant quantum computation for a web-like physical architecture of a quantum computer. Small logical units of a few qubits (realized in ion traps, for example) are linked via a photonic interconnect which provides probabilistic heralded Bell pairs [1]. Two time scales compete in this system, namely the characteristic decoherence time T_D and the typical time T_E it takes to provide a Bell pair.

Only a rare number of constructions of quantum LDPC codes are equipped with an unbounded minimum distance. Most of them are inspired by Kitaev toric codes constructed from the a tiling of the torus such as, color codes which are based on 3-colored tilings of surfaces, hyperbolic codes which are defined from hyperbolic tilings, or codes based on higher dimensional manifolds. These constructions are based on tilings of surfaces or manifolds and their parameters depend on the homology of the tiling.

In this talk, I will explain how to use lattice surgery to enact a universal set of fault-tolerant quantum operations with color codes. Along the way, I will also show how to improve existing surface-code lattice-surgery methods. Lattice-surgery methods use fewer qubits and the same time or less than associated defect-braiding methods. Per code distance, color-code lattice surgery uses approximately half the qubits and the same time or less than surface-code lattice surgery.

We consider two-dimensional lattice models that support Ising anyonic excitations and are coupled to a thermal bath, and we propose a phenomenological model to describe the resulting short-time dynamics, including pair-creation, hopping, braiding, and fusion of anyons. By explicitly constructing topological quantum error-correcting codes for this class of system, we use our thermalization model to estimate the lifetime of quantum information stored in the code space.